Mathematical Research Letters

Volume 6 (1999)

Number 2

On Ginzburg’s lagrangian construction of representations of $GL(n)$

Pages: 195 – 201

DOI: http://dx.doi.org/10.4310/MRL.1999.v6.n2.a7

Authors

Alexander Braverman (Massachusetts Institute of Technology)

Dennis Gaitsgory (Harvard University)

Abstract

In \cite{Gi} V.~Ginzburg observed that one can realize irreducible representations of the group $GL(n,{\Bbb C})$ in the cohomology of certain Springer’s fibers for the group $GL_d$ (for all $d\in {\Bbb N}$). However, Ginzburg’s construction of the action of $GL(n)$ on this cohomology was a bit artificial (he defined the action of Chevalley generators of the Lie algebra $gl_n$ on the corresponding cohomology by certain explicit correspondences, following the work of A. Beilinson, G. Lusztig and R. MacPherson (\cite{BLM}), who gave a similar construction of the quantum group $U_q(gl_n)$). In this note we give a very simple geometric definition of the action of the whole group $GL(n,{\Bbb C})$ on the above cohomology and simplify the results of \cite{Gi}.